Friday Morning. 9! 5 Nonlinear Periodic Waves in Plasma E. R. TRACY, College of William and Mary.

Transcription

1 not Friday Morning 9! 3 2F: A Generalized Energy Principle for Determining Linear and Nonlinear Stability.* P. L MORRISON,* * Institutefor Fusion Studies, The University of Texas at Austin. A generalization of the ideal MHD energy principle, 8W, is presented. The generalization is applicable to the equilibria of all of the basic nondissipative plasma models. Thus, for example, one can treat fluid equilibria with flow, models with fmite Larmour radius effects, and kinetic theories. The 6W energy principle arises because the perturbed Hamiltonian for static MHD equilibria has kinetic and potential energy terms of standard form, in which case (Liapunov) stability is determined by the potential alone. More generally the Hamiltonian siructure of plasma models in Eulerian variables is noncanonical generalized free energy (F), that has equilibria as stationary points and for which defmiteness of the second variation, 8F, is sufficient for stability This definiteness of ö2f is a more dependable criterion for practical stability than conventional linear spectral stability. Indeed, sometimes spectral theory is highly misleading because nonlinear instability for arbitrarily small perturbations can arise. This can occur when hf is indefinite, yet spectral stability theory indicates stability. Physically, 62F - the second variation of the usual energy-is the appropriate perturbed energy. Identifying the appropriate energy yields a new, more general, definition of a negative energy mode: indefiniteness of h2f and concomitant spectral stability. One can use 6F in much the same spirit as 6W; i.e. insert trial functions and then vary parameters to search for indefiniteness. A further test is used to distinguish linear instability from negative energy modes. Note that h2f is applicable even if the dielectric functional is intractable or not even defined. Many examples are available. In particular the 62F velocity thresholds for fluid and kinetic streaming instabilities are lower than those of conventional linear theory. Also MHD equilibria with flow and FLR effects 1 and the Hamiltonian is not of standard form. Nevertheless, there is a generalization of the Hamiltonian, a 3 have been treated. 2. *Suppj by the U.S. Department of Energy under Contract #DE FGO5-8OET **In collaboration with M. Kotschenreuther. 1p.J. Morrison and 3M. Greene, Phys. Rev. Lett. 45, 790 (1980). 2DD Hoim, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep. 123, 3 (1985); RD. Hazeltine, D.D. Holm, J.E. Marsden and PJ. Morrison, Proc. ICPP Lausanne 2, 204 (1984); PJ. Morrison and S. Eliezer, Phys. Rev. A 33, 4205 (1986). 3C. T. Hsu, et at., see abstract this meeting Dynamics of Resonant Magnetic Perturbations in Toroidal Plasmas with Low Collisionality.*t MICHAEL KOTSCHENREUTHER, Institutefor Fusion Studies, The University of Texas at Austin. 3. New effects considered include 1) Large self- Neoclassical effects are shown to strongly modify the dynamics of resonant magnetic perturbations, when the mean free path is long. The analysis begins by rigorously deriving reduced nonlinear fluid equations to describe the region near a rational surface using kinetic theory. Novel effects of the neoclassical terms are demonstrated both analytically and by numerical simulation of the fluid equations systematic two scale expansion in the parallel gradients, which is the kinetic analog of previous MED calculations in toroidal geometry rotational damping and bootstrap current effects arise, as previously discussed by Callen and Shaing consistent plasma currents are shown to arise in magnetic islands and stochasticity when the neoclassical transport is not intrinsically ambipolar. In stellarator fusion reactor regimes, large steady state resonant magnetic perturbations (e.g. from equilibrium Pfirsch-Schluter currents or coil errors) can be strongly reduced or healed. The linear and nonlinear stability of low and moderate m tearing and interchange modes is also affected. 2) In tokamaks, an analysis of nonlinear Rutherford Island evolution shows that bootstrap current effects contribute a destabilizing term destabilizing term dominates for island widths smaller than ajim Je. Overlap of the resulting moderate-m islands can seriously degrade confinement for3>1%. 1M. Kotschenreuther and A.Y. Aydemir, Institute for Fusion Studies Report. 2M Kotschenreuther, RD. Hazeltine and P.3. Morrison, Phys. Fluids 28, 294 (1985). 3.u. Callen and K.C. Shaing, Bull. Am. Phys. Soc. Vol. 30, 1424 (1985). 4R. Carrera, RD. Hazeltine and M. Kotschenreuther, Phys. fluids 29, 899 (1986). *In collaboration with A. Y. Aydemir twork supported by the U.S. Department of Energy. 1. These equations are derived using a 2. Strong 4. The 9! 5 Nonlinear Periodic Waves in Plasma E. R. TRACY, College of William and Mary. During the study of the nonlinear aspects of plasma behavior one often turns to simplified nodels with the hope of gaining deeper insight than can be gotten by a direct approach on the full problen. Such models retain many of the relevant physical properties of the full system, but are more amenable to study. A number of important nonlinear models (the so-called soliton systems) have the added bonus of being exactly solvable so that, in principle, we can answer rigoroisly whatever phycical guestions we wish to pose. Many scientists are familiar with soliton systems on the infinite line (i,e. when the wave disturbances are localized in space). However, the recent progress made in the study of these systems with periodic boundary conditions, which are also physically important, is much less familiar. Many important new physical affects appear and new techniqoes have been developed for their investigation. For example: the nonlinear Schrodinger egcation arises commonly in the study of modulational problems. On the infinite line this system is stable, but with periodic boundary conditions solutions exhibit instabilities. Recent arogress in the study of important nonlinear boundary value problems wifl also be discussed. Examples to be discussed include the Sinh Poisson equation the study of two oimensi nal g2iding center plasmas, and the Lioiville equation which occure in the study of ideal two dimensional ISIS eceilibria. Also to be discussed are nonintegrable models which exhibit chaotic solitons, 1 A. C. Ting, PhD. Thesis, University of Maryland (l981). 2) s. N. aian, PhD. Thesis, University of Maryland (1986). 1 which arises in *This work was carried ot in collaboration with P. H. Chen and Y. C. Lee of the University of Maryland. t::.is work was supported by NSF, UR a te Coilece Jil1Or and Mary. 1609

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